Schauder fixed point theorem
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The Schauder fixed point theorem is an extension of the Brouwer fixed point theorem to Banach spaces. It asserts that if K is a compact, convex subset of a Banach space and T is a mapping of K into itself, then T has a fixed point. It was discovered by Juliusz Schauder.
A consequence, called Schaefer's fixed point theorem, is particularly useful for proving existence of solutions to nonlinear partial differential equations. Schaefer's theorem is in fact a special case of the far reaching Leray-Schauder theorem which was discovered earlier by Schauder and Jean Leray. The statement is as follows. Let T be a continuous and compact mapping of a Banach space X into itself, such that the set
is bounded. Then T has a fixed point.
[edit] References
- D. Gilbarg, N. Trudinger Elliptic Partial Differential Equations of Second Order. ISBN 3-540-41160-7.
- L.C. Evans Partial Differential Equations.
